Question on computing direct limits Lately, I've been working on direct limits. In particular, given
$$\mathbb{Z}^n \xrightarrow{M} \mathbb{Z}^n \xrightarrow{M} \mathbb{Z}^n \xrightarrow{M} \cdots$$
where $M$ is an $n \times n$ matrix over $\mathbb{Z}$, then
(1) if the eigenvalues of $M$ are integers, the direct limit is isomorphic to the sum of $\mathbb{Z}[\frac{1}{e_i}]$'s, where $e_i$ is the $i$th eigenvalue. If $0$ is an eigenvalue, then just reduce the degree $n$ by the number of $0$ eigenvalues.
Is there a good reference to draw this conclusion from? Any book or online articles would be of great help!
(2) on the event that the eigenvalues are irrational, then the determinant is considered. For instance, the direct limit becomes $\mathbb{Z}[\frac{1}{d}]^n$, where $d$ is the determinant.
I am struggling a bit on this, and any help would be appreciated. Thanks!
 A: Let $M$ be an $R$-module and $A\in R$ not a zero divisor. Suppose we wish to compute the direct limit of the system $M\to M\to M\to\cdots$ given by repeated application of $A$. There is a nice trick that can be used to reinterpret this system as taking place entirely within the "localization" $(A)^{-1}M$:
$$\large\begin{array}{cccccccc} M & \xrightarrow{~A~} & M & \xrightarrow{~A~} & M &  \xrightarrow{~A~} & M & \xrightarrow{~A~} & \cdots \\[4pt] \uparrow & & \uparrow & & \uparrow & & \uparrow \\[4pt] M & \hookrightarrow & A^{-1}M & \hookrightarrow & A^{-2}M & \hookrightarrow & A^{-3}M & \hookrightarrow & \cdots  \end{array} $$
The up arrows represent multiplication by $A$. Check that all squares in the diagram commute. Thus the upper and lower direct systems are equivalent and their direct limits are equal.
By $(A)$ here I mean the set of all powers of $A$. Another way to think about it as the "extension of scalars" given by the tensor product $(A)^{-1}M\cong (A)^{-1}R\otimes_RM$. Note $(A)^{-1}R$ (as a ring) is the same as $R[A^{-1}]$. The direct limit, then, is $\bigcup A^{-n}M=R[A^{-1}]M=(A)^{-1}M$. An easier example (than integer matrices) is the integers by itself, two examples are covered in this question.
The rest of the exercise I assume wants Smith normal form to be used. Not sure about the ${\Bbb Z}[d^{-1}]$.
A: Let more generally $F$ be a finitely generated free module over a PID $A$ and $M : F \to F$ be an endomorphism. Then the smith normal form tells us that there is a basis of $F$ such that $M$ becomes diagonal, say $M = \mathrm{diag}(d_1,\dotsc,d_n)$ with $d_1 | \dotsc | d_n$ in $A \setminus \{0\}$. In other words, $M$ is isomorphic to the direct sum of the the endomorphisms $d_i : A \to A$. The colimit of $A \xrightarrow{d_i} A \xrightarrow{d_i} \dotsc$ is isomorphic to the localization $A_{d_i}$. Hence, the colimit of $F \xrightarrow{M} F \xrightarrow{M} $ is isomorphic to $\oplus_i A_{d_i}$.
Edit. This is not correct, since the smith normal form uses row and column operations, which means that we have to choose two bases which are different a priori. The isomorphisms above only hold when these bases are equal.
